Topic 74: Downward Spiral

Why can't we just use "ever" or "at all" in any sentence we want? What do we have to change about how a sentence works to let words like those in? In this episode, we talk about negative polarity items, or NPIs: when they can show up, why their name ismisleading, and how changing what a sentence entails changes everything for these little terms.

Quick summary:

We can't just use words like any or ever wherever we feel like; for example, a sentence like "I rode any bus out of town" is just weird. So what determines where we can put them? Well, they seem like they're okay in sentences with negation, like, "I didn't ride any bus out of town." And so terms like ever, any, at all, and idioms like budge an inch or give a hoot are known as negative polarity items, or NPIs.

But this can't be the whole story! Check out a sentence like "All the pizza that Riley ever ate before San Francisco was delicious." Now there's no negation in the sentence, and yet ever is still just fine! So what's going on? It turns out that all and not have something in common: they both create downward entailing environments. You may remember the concept of entailment from our episode about implicature, entailment, and presupposition. An entailment happens when if one sentence is true, another sentence must also be true: so like, if you've read a novel, then you've read a book.

That ordering, where novel is a subset of book, with the entailment running from the smaller to the larger set, is the way that sentences work by default; this is upward entailment. However, when you introduce not or all, the order of the entailment flips! If you haven't read a book, then you definitely couldn't have read a novel, because novels are a kind of book. Sentences where the relationship runs from the larger to the smaller set have a downward entailment. And NPIs are happy to appear so long as it's somewhere that downward entailment is happening!

Extra Materials:

In order to explain where NPIs like any and ever can go, we invoked the notion of entailment, which is a relation that can hold between two sentences, like in (1). Normally, NPIs can’t make their way into an ordinary sentence, as in (2). But when certain — often negative — words are added to those sentences, like not or without, the entailment relation that held between them gets reversed, like in (3), and NPIs can get in, as shown in (4).

(1)“Riley won the hockey game with help from her friends”ᴇɴᴛᴀɪʟꜱ

“Riley won the hockey game with help”

(2)*Riley won the hockey game with any help

(3)“Riley won the hockey game without help”ᴇɴᴛᴀɪʟꜱ

“Riley won the hockey game without help from her friends”

(4)Riley won the hockey game without any help

We had to shift our definition of entailment a little bit, to account for the fact that NPIs are sometimes restricted to only the beginnings of sentences, following quantifiers like “all” and “every.” But, on the whole, the explanation seems to work: some words create downward entailing environments, which predict where we expect to find NPIs.

Except, this can’t be right! For instance, take the sentences in (5).

(5a)Did Riley ever try broccoli pizza?

(5b)Has anyone seen Bing Bong?

NPIs, as it turns out, are actually pretty happy inside questions. But this seems to pose a big problem, since it’s hard to see how turning a sentence into a question reverses the direction of entailment. That’s because questions don’t really entail anything at all!

Remember that for one sentence to entail another, the truth of the first has to guarantee the truth of the second. But questions aren’t the sorts of things that can be either true or false. It's really weird to think of either question in (5) as having any kind of truth value, let alone either one of them entailing something else. But thankfully, the idea of either a sentence or a part of a sentence being “downward entailing” turns out just to be a specific case of a much more general phenomenon: non-veridicality.

If you know some Latin, this term basically means what it sounds like it means: a non-veridical environment is one which doesn’t guarantee the truth of its contents. So, the sentence in (6) is non-veridical, because it doesn’t guarantee that Riley likes her new school. In fact, it guarantees she doesn’t (making the sentence anti-veridical). So, negation renders any sentence it gets involved with non-veridical.

(6)Riley doesn’t like her new school

If we use this property to help describe where we find NPIs, we can not only explain why negative words open sentences up to NPIs, but why questions do as well. Since questions like the one in (5a) are often thought of by linguists as representing lists of possible answers (i.e., {Riley tried broccoli pizza, Riley didn’t try broccoli pizza}), and aren’t usually thought of by anyone as being either true or false, they fit the definition of non-veridicality.

And because non-veridicality can also explain why we find NPIs in sentences beginning with negative quantifiers like “no” and “few,” alongside yes/no questions and sentences containing full-on negation, it serves as a better predictor of the distribution of NPIs.

Now, you might’ve noticed that this idea doesn’t seem to apply to quantifiers like “every” or “all” in a straightforward way. After all, these words allow NPIs to appear only at the beginnings of sentences, but it isn’t clear how to classify only part of a sentence as non-veridical.

(7a)All of Riley’s teammates cheered for her

(7b)All of Riley’s teammates who had ever scored a goal cheered for her

Like our notion of entailment, then, we’ll need to tinker with the idea of non-veridicality a bit to get what we want. The good news is that once we do this, we’ll be left in a position to explain some things that we couldn’t quite get at before.

First, when it comes to the sets that a quantifier combines with, we can’t really say which ones are veridical (truth preserving) and which ones aren’t, since sets aren’t statements, they’re just collections of things. But, we can talk about the size of the collection; specifically, we can ask whether or not quantifiers place restrictions on the contents of their sets — like a requirement that they contain something rather than nothing.

The things is, from a purely logical standpoint, the words “every” and “all” don’t care if they combine with an empty set. And we can see this, intuitively, in (8), where the sentences seem to be true despite the fact that the set of unicorns is (sadly) empty.

(8a)Every unicorn has a single horn

(8b)All unicorns have a single horn

In this extended sense, the quantifiers “all” and “every” are non-veridical with respect to the first set they combine with, since they don’t need these sets to have anything inside them. NPIs, then, can still be connected to non-veridicality.

And this also explains why words like “both” and “each” don’t easily admit NPIs into their sentences (9), even though they seem to create downward entailing environments (10-11), which would lead us to expect the opposite.

(9) ?? Each city where Riley ever lived has a hockey league

(10a)old memories have been lostᴇɴᴛᴀɪʟꜱ

(10b)memories have been lost

(11a)each memory has been lostᴇɴᴛᴀɪʟꜱ

(11b)each old memory has been lost

These sorts of quantifiers are veridical, meaning that that they actually need the sets they combine with to have something in them. So, non-veridicality appears to serve as a worthy replacement for downward entailment, capturing in one fell swoop all the places where NPIs can go, and also where they can’t!

Discussion:

So how about it? What do you all think? Let us know below, and we’ll be happy to talk with you about the semantics of negative polarity items and where they can go. There’s a lot of interesting stuff to say, and we want to hear what interests you!